In computational anatomy
[1]}, [2]} one of the most
fundamental problems is to continuously deform an image or shape into another
and thereby obtain a natural notion of distance between them as the energy
required for such a deformation. Common methods to compute image deformations
are based on diffeomorphic deformations which assume that the images are
continuously deformed into one another with the additional property that the
inverse deformation is also continuous. This is a strong requirement for images
which implies that the 'mass' of any part of the image is conserved: we cannot
create or close 'holes'. This is also a crucial property in fluid mechanics and
in fact the theory of diffeomorphic matching carrying the moniker Large
Deformation Diffeomorphic Metric Mapping (LDDMM)
[3]}, [4]} has been inspired by fluid
mechanics. Indeed, Arnold [5]} made the central
observation that the geodesic equations for the diffeomorphism group induced by
divergence-free vector fields corresponded to that of incompressible flows. If a
strictly diffeomorphic matching is not possible or necessary, an extension of
LDDMM called metamorphosis [6]}, [7]} is
available which introduces a parameter \(\sigma ^2\)  parameterising the deviation
from diffeomorphic matching allowing for topological variations e.g. growth via
image intensity. In particular, if \(\sigma ^2=0\)  the deformation is purely
diffeomorphic as in LDDMM. See
[8]}, [9]}, [10]} for technical details
pertaining to the construction of the metamorphosis problem. While diffeomorphic
paths always exist for landmark problems [11]} this
theory allows one to match images of shapes with different topological features,
which is ill-conditioned for standard LDDMM. Indeed, even inexact matching in
LDDMM for such problems yields large energies and spurious geodesics that do not
contribute to an intuitive matching, see figure REF . As observed
here, introducing \(\sigma ^2>0\)  regularises the problem and qualitative improves
the matching.
